# If a and b are two vectors such that |a+b|=|a|, then prove that vector 2a+b is perpendicular to vector b - Mathematics

If veca and vecb are two vectors such that |veca+vecb|=|veca|, then prove that vector 2veca+vecb is perpendicular to vector vecb

#### Solution

Given |veca+vecb|=|veca|∣

|veca+vecb|^2=|veca|^2

|veca|^2+2veca.vecb+|vecb|^2=|veca|^2

2veca.vecb+|vecb|^2=0 ................(1)

Now (2veca+vecb)(vecb)=2vecavecb+vecbvecb=2vecavecb+|vecb|^2=0  using(1)

We know that, if the dot product of two vectors is zero, then either of the two vectors is zero or the vectors are perpendicular to each other.

Thus,

Given |veca+vecb|=|veca|

|veca+vecb|^2=|veca|^2

|veca|^2+2veca.vecb+|vecb|^2=|veca|^2

2veca.vecb+|vecb|^2=0 ................(1)

Now (2veca+vecb)(vecb)=2vecavecb+vecbvecb=2vecavecb+|vecb|^2=0  using(1)

We know that, if the dot product of two vectors is zero, then either of the two vectors is zero or the vectors are perpendicular to each other.

Thus, 2veca+vecb is perpendicular to vecb

Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors
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